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I'm very interested to know about historical development of power series, i.e. $$\sum_{n=0}^{\infty}a_n(x-c)^n=a_0+a_1(x-c)+a_2(x-c)^2+\dots$$ What was the situation and historical context that necessitated the discovery of power series? Why did mathematicians originally need power series?

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Some power series, like the geometric progression were indeed encountered since the ancient times, but the first person who used them systematically was I. Newton. Actually Newton considered this his main mathematical discovery: that any equation (algebraic, differential, etc.) can be solved by substituting a power series with undetermined coefficients, and then the coefficients can be found successively. This is the essential message of his two coded sentences which he sent to Leibniz via Oldenburg. You may read here a discussion about these coded messages: https://mathoverflow.net/questions/140327/arnold-on-newtons-anagram It is clear from Newton's mathematical papers, that for him "calculus" was essentially the use of power series. Mathematicians of 18s century started immediately using them for all sorts of problems. However they mostly did formal manipulations. The rigorous theory of convergence goes back to Cauchy and Abel. After that the theory essentially merged to the theory of analytic functions.

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    $\begingroup$ Concerning the coefficients being "found successfully" I suspect you really meant "found successively". $\endgroup$ – KCd Nov 23 '17 at 6:10
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As noted by Victor J. Katz, the source of Newton's introduction of infinite series is unending decimals. Victor J. Katz provides a quotation from Newton where the latter explicitly states that unending decimals were the source of his inspiration, and express wonder why nobody else tried to study these.

Unending decimals were pioneered by Simon Stevin. While Stevin is not mentioned explicitly by Newton in this quotation, decimal notation and unending decimals were by then in increasingly common use among European mathematicians, even though it took a while before they become commonplace.

Therefore the sources of infinite series can be sought in the work of Stevin, as well.

Stevin also pioneered the viewpoint of considering all numbers (rational or irrational) as being given by their unending decimal representation, in his book L'Arithmetique. He was thus a pioneer of the real number system, as well, nearly three centuries before Cantor and Dedekind.

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  • $\begingroup$ Noted by Victor J. Katz: where? $\endgroup$ – Francois Ziegler Oct 23 '17 at 3:27
  • $\begingroup$ @FrancoisZiegler, this is discussed in detail here $\endgroup$ – Mikhail Katz Oct 23 '17 at 11:24
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in his first paper on the calculus (1669) Newton used the series to integrate the function $y=\frac1{1+x^2}$ where he used the binomial theorem and integrated term by term the produced the infinite polynomial.

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